Problem: Let N be a positive odd number.

There are N coins, numbered 1, 2, \ldots, N. For each i (1 \leq i \leq N), when Coin i is tossed, it comes up heads with probability p_i and tails with probability 1 - p_i.

Taro has tossed all the N coins. Find the probability of having more heads than tails.

Constraints

* N is an odd number.
* 1 \leq N \leq 2999
* p_i is a real number and has two decimal places.
* 0 < p_i < 1

Input

Input is given from Standard Input in the following format:


N
p_1 p_2 \ldots p_N


Output

Print the probability of having more heads than tails. The output is considered correct when the absolute error is not greater than 10^{-9}.

Examples

Input

3
0.30 0.60 0.80


Output

0.612


Input

1
0.50


Output

0.5


Input

5
0.42 0.01 0.42 0.99 0.42


Output

0.3821815872

Solution (in JAVA):

Solution:
/******************************************************************************

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                Code, Compile, Run and Debug java program online.
Write your code in this editor and press "Run" button to execute it.

*******************************************************************************/
import java.io.*;
import java.util.*;
public class Main
{
    public static double func(double p [], int n , int tail,int curr,double dp[][])
    {
        
        
        
        if((n-tail)<=tail)
        {
            return 0.0;
        }
        
        if(curr>=n)
        {
            
            return 1.0;
            
        }
        if(dp[curr][n-tail-1]>-1.0)return dp[curr][n-tail-1];
        return dp[curr][n-tail-1]=p[curr]*func(p,n,tail,curr+1,dp)+(1.0-p[curr])*func(p,n,tail+1,curr+1,dp);
      
    }
    
    
	public static void main(String[] args) throws Exception {
	
		BufferedReader br =new BufferedReader(new InputStreamReader(System.in));
		int n=Integer.parseInt(br.readLine());
		String ss[]=(br.readLine()).split(" ");
		
		double p[]=new double [n];
		for (int i=0;i<n;i++)
		{
		    p[i]=Double.parseDouble(ss[i]);
		    
		}
	    double dp[][]=new double[n][n];
	    for(double row[]:dp)
	    Arrays.fill(row,-1.0);
	    
		double ans1=func(p,n,0,0,dp);
		System.out.println(ans1);
		
		
		
		
	}
}